Before you even look at the repertoire on Viktoria Mullova's latest recording, the title brings you up short. The Peasant Girl? Referring to one of the world's most refined and glamorous classical musicians?

As Mullova tells it, however, it's a perfect fit. "My husband and a friend were joking with me one day, and they said, 'You think you can play Beethoven and Bach, but you're a peasant!' My family came from a village in Ukraine, and my grandmother didn't know how to read." While bantering, the three were listening to a jazz-rock track by Weather Report, which Mullova's husband, cellist Matthew Barley, was thinking of arranging for an album. "We knew it only by its track number. Suddenly we saw the CD – and what was it called? The Peasant!"

The Peasant Girl, which forms the basis of the programme of Mullova's late-night Prom on Thursday, draws from wide-ranging spheres – violin-cello duos by Bartók and Kodály rub shoulders with Barley's arrangements of Youssou N'Dour and French Gypsy-music band Bratsch. There is also an original composition by Barley himself – Yura, named after Mullova's father. It's this haunting piece that emerges as the heart of the album, and talking to Mullova it becomes clear why.

She has been based for the last two decades in London, where she, Barley and her three children live in a house in Holland Park, built to their own design. From Ukraine to that house is quite a leap in only two generations, as Mullova acknowledges. "But one good thing about the Soviet Union was that people from the villages were given the opportunity to go to university. My mother and father both did, and were given good jobs. They wanted their children to be educated." She was started on the violin aged four, practising every day with her father, an aeroplane engineer. "It was so hard to play," she remembers. "And I didn't like carrying it around – people would see that I was different. At the same time, I knew that I was good at it. I became as ambitious as my parents were. But practising was not a good thing." When did that change? "It's still that way!" She rubs her shoulder. "Physically, it's hard work. It's good when it goes well – but that's rare."

Yet practising got her into the Moscow Conservatoire, where foreign students started to pique an interest in new musical worlds. When she won the Sibelius competition in Helsinki, she gave some of her prize money to students at the academy there to buy her some LPs. "They gave me the Bee Gees, Barbra Streisand, Pink Floyd, Herbie Hancock, Santana. We couldn't buy those in Russia, and all the names were new to me. I mean, it was 1980 and I hadn't even heard of the Beatles."

In 1982 she defected to the US, driving across the border from Finland to Sweden while on tour with her then boyfriend. They arrived in Stockholm, disguised in blond wigs, only to find the US embassy closed for the weekend of 4 July, and had to hide out in a hotel for two days, not leaving their room even to get food.

"I was determined to leave Russia," says Mullova. "I hated it so much that for many years I denied everything Russian. Later on I started to miss my parents – but then Perestroika started. I first went back in 1991." Did they understand why she left? "Of course. But my parents had huge problems [from the authorities] at their work. And both my sisters left Russia too."

Mullova's father died last summer, the day after the piece Barley had written was first performed. "He had cancer, and in the last few years we had become so close. I went to Russia every month to see him, and we spoke every day on Skype."

Then, last summer, she went to meet his relatives in Irkutsk. "It's very far away, near Mongolia. Suddenly I was full of Russia! We went to Lake Baikal for eight days, collecting mushrooms, fishing, singing songs by the fire. My children came too, and Matthew – that's where Yura was composed. My father couldn't come because he already couldn't walk, and two months later he died. It was the most extraordinary time – sad but, at the same time, so much happiness. It was a rediscovery of myself."

Had Mullova remained within the rigid Soviet system, it's unlikely her career would have branched out as it has. In recent years she has become known as much for her Bach and Vivaldi as for her Sibelius. "I'm lucky that I found my way to the baroque world. Then my classical stuff started to change too – I started to play Beethoven and Brahms concertos on gut strings ."

Then there's the music that she and Barley make together. In 2000 they recorded Through the Looking Glass, a reimagining of music by artists including Miles Davis, the Bee Gees and, again, Weather Report, and Mullova has other non-classical schemes brewing.

But she is far from neglecting the classical repertoire; last week she was at the Salzburg festival playing Beethoven sonatas. And then there's the new Violin and Cello Concerto by Thomas Larcher, which she and Barley will be premiering with the BBC Scottish Symphony Orchestra in Thursday's early-evening Prom. When we speak, she has yet to hear it with the orchestra. But, she says, it has "beautiful, tender" melodies. As it turns out, Larcher developed one of those from a tune he heard played by Gypsy buskers on the Berlin underground. For Mullova, it's another connection with the Gypsy world. But then she was probably always more of a Gypsy than a peasant – and now she has found a home.

• Viktoria Mullova and Matthew Barley perform in the late-night Prom 46 on 18 August; Mullova performs Brahms's Violin Concerto with the Philharmonia Orchestra at the Royal Festival Hall on 25 September.